Free math worksheets

- What does the exponent tell you about repeated multiplication? - Count how many factors of \(10\) you need. - Multiply the factors after writing them out.

1. The exponent \(4\) tells how many factors of \(10\) are multiplied: \(10^4 = 10 \times 10 \times 10 \times 10\). 2. The product is \(10{,}000\).

\(10^4 = 10 \times 10 \times 10 \times 10 = 10{,}000\)

- Rewrite all values in the same form. - Compare the number of digits or the exponents of \(10\). - With the same positive coefficient, a greater exponent gives a greater value.

1. Write each number in standard form: \(700\), \(7000\), \(70{,}000\), \(7{,}000{,}000\), and \(70{,}000{,}000\). 2. Compare the place values: \(700 < 7000 < 70{,}000 < 7{,}000{,}000 < 70{,}000{,}000\).

\(700 < 7 \times 10^3 < 7 \times 10^4 < 7 \times 10^6 < 70{,}000{,}000\)

- Write each power of \(10\) in standard form. - Multiplying by \(10\) makes each digit's value \(10\) times as large; use that place-value relationship for \(10^n\). - Check the number of zeros in each result.

1. a) \(5 \times 10^4 = 50{,}000\). 2. b) \(23 \times 10^6 = 23{,}000{,}000\). 3. c) \(10^8 = 100{,}000{,}000\). 4. d) \(402 \times 10^3 = 402{,}000\).

a) \(50{,}000\) b) \(23{,}000{,}000\) c) \(100{,}000{,}000\) d) \(402{,}000\)

- Rewrite both sides in standard form. - Compare place values after converting. - Check whether the powers of \(10\) represent the same number of zeros.

1. a) \(6 \times 10^4 = 60{,}000\), so \(60{,}000 > 6000\). 2. b) \(2 \times 10^5 = 200{,}000\), so the values are equal. 3. c) \(9 \times 10^2 = 900\) and \(9 \times 10^3 = 9000\), so \(900 < 9000\). 4. d) \(3 \times 10^4 = 30{,}000\), so \(40{,}000 > 30{,}000\).

a) \(6 \times 10^4 > 6000\) b) \(200{,}000 = 2 \times 10^5\) c) \(9 \times 10^2 < 9 \times 10^3\) d) \(40{,}000 > 3 \times 10^4\)

- Relate each exponent to the number of place-value shifts. - Write the power of \(10\) in standard form. - Multiply by the coefficient when one is given.

1. a) Since \(3 \times 10^4 = 30{,}000\), \(n = 4\). 2. b) \(10^4 = 10{,}000\), so \(x = 5 \times 10{,}000 = 50{,}000\). 3. c) \(10^5 = 100{,}000\), so \(n = 5\). 4. d) \(10^1 = 10\), so \(x = 8 \times 10 = 80\).

a) \(n = 4\) b) \(x = 50{,}000\) c) \(n = 5\) d) \(x = 80\)

- Recall how many zeros are in a million, billion, and trillion. - Use the exponent to determine the place-value shift. - For part b, include as many trailing zeros as possible in the power of \(10\).

1. a) \(8 \times 10^5 = 800{,}000\), which is eight hundred thousand. 2. b) \(400{,}000{,}000 = 4 \times 10^8\). 3. c) Six trillion is \(6{,}000{,}000{,}000{,}000\). 4. d) \(15 \times 10^4 = 15 \times 10{,}000 = 150{,}000\).

a) eight hundred thousand b) \(4 \times 10^8\) c) \(6{,}000{,}000{,}000{,}000\) d) \(150{,}000\)

- Write each number in standard form first. - Count the trailing zeros. - Include as many trailing zeros as possible in the power of \(10\).

1. a) Eighty billion is \(80{,}000{,}000{,}000 = 8 \times 10^{10}\). 2. b) Five hundred million is \(500{,}000{,}000 = 5 \times 10^8\). 3. c) Two thousand is \(2000 = 2 \times 10^3\). 4. d) Seven trillion is \(7{,}000{,}000{,}000{,}000 = 7 \times 10^{12}\).

a) \(8 \times 10^{10}\) b) \(5 \times 10^8\) c) \(2 \times 10^3\) d) \(7 \times 10^{12}\)

- The exponent in a power of ten matches the number of zeros in that place value. - Match each digit to its place value. - Omit a term when the digit in that place is \(0\).

1. a) The place-value expansion is \(9000+300+80+2\), so \(9382=9\times10^3+3\times10^2+8\times10+2\). 2. b) The nonzero place values are \(50{,}000\), \(400\), \(10\), and \(7\), so \(50{,}417=5\times10^4+4\times10^2+1\times10+7\). 3. c) The nonzero place values are \(600{,}000\), \(3000\), and \(9\), so \(603{,}009=6\times10^5+3\times10^3+9\).

a) \(9\times10^3+3\times10^2+8\times10+2\) b) \(5\times10^4+4\times10^2+1\times10+7\) c) \(6\times10^5+3\times10^3+9\)

- Identify the place value of each digit. - The exponent in \(10^n\) tells how many factors of \(10\) are multiplied. - Include zeros in standard form for place values that are missing from the sum. - How many zeros are in \(10^7\)?

1. a) Match every digit with its place value: \(60{,}403 = 6 \times 10^4 + 0 \times 10^3 + 4 \times 10^2 + 0 \times 10^1 + 3 \times 10^0\). 2. b) Match every digit with its place value: \(2{,}008{,}000 = 2 \times 10^6 + 0 \times 10^5 + 0 \times 10^4 + 8 \times 10^3 + 0 \times 10^2 + 0 \times 10^1 + 0 \times 10^0\). 3. \(5 \times 10^7 = 50{,}000{,}000\), \(2 \times 10^5 = 200{,}000\), and \(9 \times 10^1 = 90\). Their sum is \(50{,}200{,}090\).

1. a) \(6 \times 10^4 + 0 \times 10^3 + 4 \times 10^2 + 0 \times 10^1 + 3 \times 10^0\) 1. b) \(2 \times 10^6 + 0 \times 10^5 + 0 \times 10^4 + 8 \times 10^3 + 0 \times 10^2 + 0 \times 10^1 + 0 \times 10^0\) 2. \(50{,}200{,}090\)

- The exponent records the number of equal factors. - Here every factor is \(10\). - After writing the power, evaluate it in standard form.

1. Six factors of \(10\) are written as \(10^6\). 2. \(10^6 = 1{,}000{,}000\).

a) \(10^6\) b) \(1{,}000{,}000\)

- Rewrite both sides in the same form. - Count place-value shifts carefully. - Recall that one billion is \(10^9\).

1. a) \(2 \times 10^6 = 2{,}000{,}000\), which is less than \(20{,}000{,}000\). 2. b) \(10^3 \times 10^2 = 1000 \times 100 = 100{,}000 = 10^5\). 3. c) \(500 \times 10^3 = 500{,}000\), and \(5 \times 10^5 = 500{,}000\). 4. d) One billion is \(1{,}000{,}000{,}000 = 10^9\).

a) \(<\) b) \(=\) c) \(=\) d) \(=\)

- Recall the place values for million and billion. - Count the trailing zeros in standard form. - Use the greatest possible power of \(10\) in the last column.

1. Four million is \(4{,}000{,}000 = 4 \times 10^6\). 2. \(60{,}000{,}000{,}000\) is sixty billion, and \(60{,}000{,}000{,}000 = 6 \times 10^{10}\). 3. Three billion is \(3{,}000{,}000{,}000 = 3 \times 10^9\).

<table><tr><th>Number in words</th><th>Standard form</th><th>Using a power of \(10\)</th></tr><tr><td>nine hundred thousand</td><td>\(900{,}000\)</td><td>\(9 \times 10^5\)</td></tr><tr><td>four million</td><td>\(4{,}000{,}000\)</td><td>\(4 \times 10^6\)</td></tr><tr><td>sixty billion</td><td>\(60{,}000{,}000{,}000\)</td><td>\(6 \times 10^{10}\)</td></tr><tr><td>three billion</td><td>\(3{,}000{,}000{,}000\)</td><td>\(3 \times 10^9\)</td></tr></table>

- Convert each number word to standard form first. - Count the trailing zeros to determine the exponent. - Make sure the coefficient has no trailing zeros.

1. a) Twenty-five trillion is \(25 \times 10^{12}\). 2. b) Five hundred ten million is \(510 \times 10^6\). Removing the trailing zero from the coefficient gives \(51 \times 10^7\). 3. c) Forty billion is \(40 \times 10^9\). Removing the trailing zero from the coefficient gives \(4 \times 10^{10}\).

a) \(25 \times 10^{12}\) b) \(51 \times 10^7\) c) \(4 \times 10^{10}\)

- The exponent tells how many factors of \(10\) are multiplied together. - Count the trailing zeros to find the greatest power of ten that is a factor. - In part b), evaluate each product before adding.

1. a) The number \(19{,}000{,}000\) has six trailing zeros, so \(19{,}000{,}000=19\times10^6\). The number \(800{,}000{,}000{,}000\) has eleven trailing zeros, so \(800{,}000{,}000{,}000=8\times10^{11}\). 2. b) Evaluate the powers: \(10^4=10{,}000\) and \(10^2=100\). 3. Multiply: \(3\times10{,}000=30{,}000\) and \(7\times100=700\). 4. Add: \(30{,}000+700=30{,}700\).

a) \(19\times10^6\) and \(8\times10^{11}\) b) \(30{,}700\)

- Count the zeros in each number. - Relate the number of zeros to the exponent in a power of \(10\). - Evaluate each term in the sum before adding. - In part 3, move all trailing zeros into the power of \(10\).

1. Count the zeros in each power of \(10\): \(100{,}000 = 10^5\), \(10{,}000{,}000 = 10^7\), and \(1{,}000{,}000{,}000 = 10^9\). 2. \(7 \times 10^4 = 70{,}000\) and \(3 \times 10^2 = 300\). Therefore, \(70{,}000 + 300 = 70{,}300\). 3. Since \(150\) million kilometers is \(150{,}000{,}000\,\text{km}\), removing the trailing zeros gives \(150{,}000{,}000 = 15 \times 10^7\).

1. \(10^5\), \(10^7\), and \(10^9\). 2. \(70{,}300\). 3. \(15 \times 10^7\,\text{km}\).

- Evaluate each power-of-ten product first. - Align equal place values when adding. - A missing power of ten means the digit in that place is \(0\). - Compare the number of digits before comparing individual digits.

1. Evaluate each term: \(4\times10^5=400{,}000\), \(2\times10^3=2000\), and \(7\times10=70\). 2. Add: \(400{,}000+2000+70+5=402{,}075\). The ten-thousands and hundreds digits are \(0\). 3. The number \(402{,}075\) has six digits, while \(42{,}705\) has five digits, so \(402{,}075>42{,}705\).

a) \(402{,}075\) b) \(402{,}075>42{,}705\)